First passages in bounded domains: when is the mean first passage time meaningful?

Topography of chance

We present a model of multiplicative Langevin dynamics that is based on two foundations: the Langevin equation and the notion of multiplicative evolution. The model is a nonlinear mechanism transforming a white-noise input to a dynamic-equilibrium output, using a single control: an underlying convex U-shaped potential function. The output is quantified by a stationary density which can attain a given number of shapes and a given number of randomness categories. The model generates each admissible combination of the output's shape and randomness in a universal and robust fashion. Moreover, practically all the probability distributions that are supported on the positive half-line, and that are commonly encountered and applied across the sciences, can be reverse engineered by this model. Hence, this model is a universal equilibrium mechanism, in the context of multiplicative dynamics, for the robust generation of "chance": the model's output. In turn, the properties of the produced "chance," the output's shape and randomness, are determined with mathematical precision by the control's landscape, its topography. Thus, a topographic map of chance is established. As a particular application, probability distributions with power-law tails are shown to be universally and robustly generated by controls on the "edge of convexity": convex U-shaped potential functions with asymptotically linear wings.

Search times with arbitrary detection constraints

Random encounters in space are central to describing diffusion-limited reactions, animal foraging, search processes, and many other situations in nature. These encounters, however, are often constrained by the capacity of the searcher to detect and/or recognize its target. This can be due to limited binding and perception abilities of the searcher or hiding and avoiding mechanisms used by the target. Hence detection failure upon passage over the target location turns the process into an n-passage problem, with n being random. Here we provide a general description of this detection problem for arbitrary dimensions and arbitrary detection constraints. The mean detection time (MDT) for a random searcher embedded in a sea of homogeneously distributed targets is obtained as a function of the target density ρ, the size domain L, and the effective detection distance a. While the scaling with ρ and L is found to be universal and equivalent to that found for the corresponding first-passage problem, the scaling of the MDT on a depends on the specific detection mechanism considered.

First-passage properties of molecular spiders

Molecular spiders are synthetic catalytic DNA-based nanoscale walkers. We study the mean first-passage time for abstract models of spiders moving on a finite two-dimensional lattice with various boundary conditions and compare it with the mean first-passage time of spiders moving on a one-dimensional track. We evaluate by how much the slowdown on newly visited sites, owing to catalysis, can improve the mean first-passage time of spiders and show that in one dimension, when both ends of the track are an absorbing boundary, the performance gain is lower than in two dimensions, when the absorbing boundary is a circle; this persists even when the absorbing boundary is a single site.

Uniform asymptotic approximation of diffusion to a small target

The problem of the time required for a diffusing molecule, within a large bounded domain, to first locate a small target is prevalent in biological modeling. Here we study this problem for a small spherical target. We develop uniform in time asymptotic expansions in the target radius of the solution to the corresponding diffusion equation. Our approach is based on combining expansions of a long-time approximation of the solution, involving the first eigenvalue and eigenfunction of the Laplacian, with expansions of a short-time correction calculated by a pseudopotential approximation. These expansions allow the calculation of corresponding expansions of the first passage time density for the diffusing molecule to find the target. We demonstrate the accuracy of our method in approximating the first passage time density and related statistics for the spherically symmetric problem where the domain is a large concentric sphere about a small target centered at the origin.

Size independence of statistics for boundary collisions of random walks and its implications for spin-polarized gases

A bounded random walk exhibits strong correlations between collisions with a boundary. For a one-dimensional walk, we obtain the full statistical distribution of the number of such collisions in a time t. In the large t limit, the fluctuations in the number of collisions are found to be size independent (independent of the distance between boundaries). This occurs for any interboundary distance, from less to greater than the mean free path, and means that this boundary effect does not decay with increasing system size. As an application, we consider spin-polarized gases, such as 3-helium, in the three-dimensional diffusive regime. The above results mean that the depolarizing effect of rare magnetic impurities in the container walls is orders of magnitude larger than a Smoluchowski assumption (to neglect correlations) would imply. This could explain why depolarization is so sensitive to the container's treatment with magnetic fields prior to its use.

Assessing the inherent uncertainty of one-dimensional diffusions

In this paper we assess the inherent uncertainty of one-dimensional diffusion processes via a stochasticity classification which provides an à la Mandelbrot categorization into five states of uncertainty: infra-mild, mild, borderline, wild, and ultra-wild. Two settings are considered. (i) Stopped diffusions: the diffusion initiates from a high level and is stopped once it first reaches a low level; in this setting we analyze the inherent uncertainty of the diffusion's maximal exceedance above its initial high level. (ii) Stationary diffusions: the diffusion is in dynamical statistical equilibrium; in this setting we analyze the inherent uncertainty of the diffusion's equilibrium level. In both settings general closed-form analytic results are established, and their application is exemplified by stock prices in the stopped-diffusions setting, and by interest rates in the stationary-diffusions setting. These results provide a highly implementable decision-making tool for the classification of uncertainty in the context of one-dimensional diffusions.